Example Pipe Flow of Newtonian Fluid

In pipe flow, the fluid moves as a result of a pressure gradient along the pipe see Fig. 5.9. 

The velocity at the wall is zero and maximum at the center. The velocity gradient (shear rate) is zero in the center and maximum at the wall. The momentum balance for this case can be determined by taking a force balance on a small fluid element as shown in Fig. 5.9 this gives  

This same result can be obtained directly from momentum balance Eq. 5.4(c) by eliminating the stress derivatives in the tangential (0) and axial direction (z) this yields  

Verify that Eqs. 2 and 3 are the same when x/r = dx/dr, which is true in this case. For a Newtonian fluid, the shear stress can be related to the velocity gradient by  

Since the velocity gradient is zero at the center (r = 0), the integration constant C, has to be zero. The velocity profile is obtained by integrating Eq. 6  

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